The Democratic and Republican parties did not suddenly “split” from one another. The modern two-party system in the United States took shape over time, with the Republican Party founded in 1854 and the Democratic Party tracing back to earlier political factions that became organized in the late 1820s.
How it happened
In the earliest years of the United States, politics was first dominated by a different rivalry: the Federalists versus the Democratic-Republicans. The Democratic-Republican Party formed in the early 1790s around Thomas Jefferson and James Madison, largely in opposition to the Federalists’ stronger central government.
That older system broke apart in the 1820s. After the election of 1824, the Democratic-Republicans fractured into competing camps, and one of those camps became the Democratic Party under Andrew Jackson. This is why the Democratic Party is usually treated as the older of the two modern parties, even though its ideology and coalitions changed over time.
Where Republicans came from
The Republican Party emerged much later, in 1854, during the crisis over slavery expansion. It grew quickly by bringing together anti-slavery voters and other groups that opposed the existing political order. By the election of 1860, the Republican Party had become a major national force.
The short answer
If the question is when the two parties became the main rivals in U.S. politics, the best answer is the 1850s. If the question is when their older roots began to separate, the split reaches back to the 1820s, with the Democratic Party emerging from the collapse of the Democratic-Republicans and the Republican Party forming later in 1854.
So the two parties did not split from each other as one old party breaking neatly in half. They came out of different political realignments, and the modern Democratic-Republican-to-Democratic line, plus the later rise of the Republican Party, created the two-party system Americans recognize today.